Skip to main content

Exact and numerical solutions for the nanosoliton of ionic waves propagating through microtubules in living cells


In this article, the Paul–Painleve approach (PPA) which was formulated recently and built on the balance role has been used perfectly to achieve new impressive solitary wave solutions to the nanosoliton of ionic waves (NSOIW) propagating along the microtubules in the living cells. In addition, variational iteration method (VIM) has been applied in the same vein and parallel to establish numerical solutions of this model.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    N A Kudryashov, Optik. 183, 642 (2019)

    ADS  Article  Google Scholar 

  2. 2.

    H Fröhlich, Int. J. Quantum Chem. 2641, 153 (1968)

    Google Scholar 

  3. 3.

    M V Sataric, J A Tuszyosky and R B Zakula, Phys. Rev. E 48, 589 (1993)

    ADS  Article  Google Scholar 

  4. 4.

    M V Sataric and J A Tuszynski, J. Biol. Phys. 31, 487 (2005)

    Article  Google Scholar 

  5. 5.

    M V Sataric, M Dragic and D Sejulic, Rom. Rep. Phys. 63, 624 (2011)

  6. 6.

    A M Wazwaz, Appl. Math. Comput. 212, 120 (2009)

    MathSciNet  Google Scholar 

  7. 7.

    S S Ray and A K Gupta, J. Math. Chem. 52, 1066 (2014)

    MathSciNet  Article  Google Scholar 

  8. 8.

    S S Ray, Numer. Meth. Partial Diff. Eqs 37, 341 (2021)

    Article  Google Scholar 

  9. 9.

    N Raza, M S Osman, A-H Abdel-Aty and H R Besbes, Adv. Diff. Eqs 2020, 517 (2020)

    Article  Google Scholar 

  10. 10.

    N Raza, S Arshed and A Javid, Int. J. Nonlinear Sci. Numer. Simul. 21, 855 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Md A Kayum, S Ara, H K Barman and M Ali Akbar, Results Phys. 18, 103269 (2020)

  12. 12.

    H K Barman, A R Seadawy, M Ali Akbar and D Baleanu, Results Phys. 17, 103131 (2020)

    Article  Google Scholar 

  13. 13.

    H M E Zahran, World J. Nano Sci. Eng. 5, 78 (2015)

    ADS  Article  Google Scholar 

  14. 14.

    H M E Zahran, Adv. Nanopart. 4, 25 (2015)

    Article  Google Scholar 

  15. 15.

    A H Abdel Aty, M A Khater, R A M Attia and H Eleuch, Mathematics 8, 697 (2020)

Download references

Author information



Corresponding author

Correspondence to Ahmet Bekir.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bekir, A., Zahran, E.H.M. Exact and numerical solutions for the nanosoliton of ionic waves propagating through microtubules in living cells. Pramana - J Phys 95, 158 (2021).

Download citation


  • Painleve approach
  • nanosoliton of ionic wave model
  • variational iteration method
  • travelling wave solutions
  • numerical solutions


  • 02.60.Cb
  • 04.20.Jb
  • 05.45.Yv